How do I know whether a probability model is uniform?

Compare the probabilities of the outcomes you listed. If every listed outcome has the same probability, the model is uniform. If any two differ, it is non-uniform. Be careful: what looks like a neat list of outcomes is often non-uniform, as with the sums 2 through 12 from two number cubes.

Why must probabilities add to exactly 1?

Because your list of outcomes covers everything that can happen, and something must happen. A sum less than 1 means you left out an outcome; a sum greater than 1 usually means outcomes overlap or a probability is too large. Checking the sum is the fastest way to catch those errors.

How many trials should a simulation have?

More trials give a more reliable estimate. Twenty is a workable minimum for classwork, and 50 or 100 is much better if you can pool results with classmates. Decide the number before you start, and always report the number of trials with your estimate.

What if my simulation answer does not match the exact probability?

That is expected. A simulation estimates a probability, so results vary from run to run. Your design is wrong only if the device's probabilities do not match the real situation, if one trial does not model the whole event, or if you divided by the number of digits instead of the number of trials.

M7MATH-10.3

Probability Models & Simulations

Learn to build uniform and non-uniform probability models, check that all probabilities add to 1, and design simulations that estimate tricky probabilities.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Probability Models & Simulations, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Some probabilities are easy to reason out: a fair number cube lands on 4 with probability

\frac{1}{6}

. But what about the chance that a basketball player who makes 60 percent of her free throws makes at least two of her next three? Or the chance that you collect all four prize figures when each cereal box has one at random? Those questions are awkward to compute directly in Grade 7 — but you can still get a very good estimate by imitating the situation with random digits, spinners, or coins.

This lesson has two connected halves. First you build a probability model: a complete list of outcomes, each paired with its probability, adding to exactly 1. Then you use that model to design a simulation whose trials stand in for the real event. Run enough trials, count the successes, and the relative frequency becomes your estimate.

Probability Models: Uniform and Non-Uniform

A probability model is a list of every possible outcome of a chance situation, with a probability attached to each one. Two rules make a model legal: every probability is between 0 and 1, and all the probabilities together sum to exactly 1.

A model is uniform when all outcomes have the same probability. Rolling a fair number cube gives six outcomes each with probability

\frac{1}{6}

, and

6 \times \frac{1}{6} = 1

. Drawing one card at random from a shuffled set of 20 index cards is uniform too, with each card at

\frac{1}{20}

.

A model is non-uniform when the outcomes are not equally likely. A bag with 3 red, 1 blue, and 6 green marbles gives

P(\text{red}) = \frac{3}{10}

P(\text{blue}) = \frac{1}{10}

P(\text{green}) = \frac{6}{10}

. Still sums to 1, still a valid model — just not uniform.

Situation	Outcomes	Type
Fair coin flip	H, T at $\frac{1}{2}$ each	Uniform
Spinner with halves red, quarter blue, quarter yellow	$\frac{1}{2}, \frac{1}{4}, \frac{1}{4}$	Non-uniform
Sum of two number cubes	2 through 12	Non-uniform

The biggest error students make is assuming uniformity because the outcomes are listed neatly. The sum of two number cubes has 11 outcomes, but they are not

\frac{1}{11}

each: a sum of 7 happens in six of the 36 equally likely pairs, while a sum of 2 happens in only one. Ask yourself what the equally likely pieces really are before assigning probabilities.

Building a Model and Checking the Sum

To build a model, work in a fixed order so you do not skip or double-count outcomes. List the outcomes, count how many equally likely ways produce each one, then write each probability as a fraction of the total.

Suppose a spinner has 8 equal sectors: 4 labeled A, 3 labeled B, 1 labeled C. The eight sectors are the equally likely pieces, so

P(A) = \frac{4}{8}

P(B) = \frac{3}{8}

P(C) = \frac{1}{8}

. Check:

\frac{4}{8} + \frac{3}{8} + \frac{1}{8} = \frac{8}{8} = 1

. That check is not busywork — it catches missing outcomes. If your probabilities sum to

\frac{7}{8}

, you forgot a sector.

The sum rule also lets you find a missing probability. If a bookstore's model for the number of books a customer buys is

P(0) = 0.20

P(1) = 0.35

P(2) = 0.30

, and

P(3 \text{ or more})

is unknown, then

P(3 \text{ or more}) = 1 - (0.20 + 0.35 + 0.30) = 0.15

Two cautions. First, outcomes in a model must not overlap. A model listing "even" and "greater than 3" for a number cube is broken, because 4 and 6 belong to both. Second, decimals and fractions must be handled carefully:

0.2 + 0.3 + 0.4 = 0.9

, not 1, so that model is incomplete even though it looks tidy. Always write the sum out and confirm it equals 1 before you use a model for anything else.

Designing a Simulation

A simulation replaces a real experiment with a chance device that has the same probability model. You use one when the real thing is too slow, too expensive, or too hard to compute — like the chance of getting at least three heads in five flips, or of a family with three children having all girls.

A good design answers four questions in writing before any trials happen.

Step	Question to answer
1	What device matches the probabilities?
2	How do outcomes map to the device?
3	What counts as one trial?
4	How many trials, and what am I counting?

For a 60 percent free-throw shooter, random digits 0 through 9 work well: let 0, 1, 2, 3, 4, 5 mean "make" (six of ten digits, so probability 0.6) and 6, 7, 8, 9 mean "miss". For a

\frac{1}{3}

chance, digits are awkward — 9 does not divide by 10 evenly — so use a number cube with 1 and 2 meaning success, or spin a three-part spinner instead.

One trial must model the whole event you care about, not a single piece of it. If the question is about three free throws, one trial is a group of three digits, and one trial produces one success-or-failure result. Then

P(\text{event}) \approx \frac{\text{number of successful trials}}{\text{total trials}}

The most common design error is letting the device's probabilities drift from the real ones — for instance using a coin for a 60 percent shooter because heads and tails are "close enough." They are not; a coin models 50 percent. Match the numbers exactly, or the estimate answers a different question.

Reading Simulation Results Honestly

A simulation gives an estimate, not a truth. Twenty trials of the free-throw situation might give 13 successes, suggesting 0.65, while another twenty give 15, suggesting 0.75. Both are legitimate results from the same correct design. As the number of trials grows, the relative frequency tends to settle near the true probability — this is the same long-run idea you met when comparing experimental and theoretical probability.

So when you report a simulation result, report the number of trials with it. "About 0.70, based on 50 trials" is a complete answer; "0.70" alone hides how much confidence it deserves. If a classmate's estimate differs from yours, the first question is not "who is wrong" but "how many trials did each of us run, and did we model the same event?"

Three places students go wrong when interpreting results:

Counting digits instead of trials. If 60 digits were used in 20 trials of three shots, the denominator is 20, not 60.

Stopping early because the answer "looks right." Deciding to quit after a run of successes biases the estimate upward. Fix the number of trials in advance.

Treating the estimate as exact. If a simulation gives

\frac{14}{20} = 0.70

and the true value is 0.648, the simulation is not broken; it simply has ordinary variability.

One more habit worth building: after running the simulation, ask whether the answer is reasonable. A probability estimate above 1, or a "chance of at least two makes" smaller than the "chance of all three makes," signals a counting mistake rather than bad luck.

Key terms

Probability model.: A complete list of all outcomes of a chance situation, each paired with a probability, where the probabilities sum to 1.
Uniform probability model.: A model in which every outcome has the same probability, such as $\frac{1}{6}$ for each face of a fair number cube.
Non-uniform probability model.: A model in which outcomes have different probabilities, such as a spinner with unequal sectors or the sum of two number cubes.
Simulation.: An experiment using a chance device (digits, cubes, coins, spinners) whose probabilities match a real situation, used to estimate a probability.
Trial.: One complete repetition of a simulation that models the whole event of interest and produces one success-or-failure result.
Relative frequency.: The number of successful trials divided by the total number of trials; it estimates the probability of an event.
Outcome.: A single possible result of a chance situation; outcomes in a model must not overlap and must cover every possibility.

Worked example

Jada makes 60 percent of her free throws. Design and run a simulation to estimate the probability that she makes at least 2 of her next 3 free throws. Use these random digits, read left to right: 4 7 1 9 0 3 6 6 5 2 8 4 1 7 0 9 3 2 5 6 8 1 4 7 2 0 6 3 9 5.

Step 1: Choose a device. Random digits 0 through 9 work because 60 percent equals six out of ten digits.

Step 2: Assign outcomes. Digits 0, 1, 2, 3, 4, 5 mean make (probability 0.6). Digits 6, 7, 8, 9 mean miss (probability 0.4). Check the model:

0.6 + 0.4 = 1

, so it is valid.

Step 3: Define one trial. One trial is a group of 3 digits, since the question is about 3 free throws. A trial is a success if 2 or 3 of the digits are makes.

Step 4: Run the trials. Thirty digits give 10 trials.

Trial	Digits	Makes	Success?
1	4 7 1	2	Yes
2	9 0 3	2	Yes
3	6 6 5	1	No
4	2 8 4	2	Yes
5	1 7 0	2	Yes
6	9 3 2	2	Yes
7	5 6 8	1	No
8	1 4 7	2	Yes
9	2 0 6	2	Yes
10	3 9 5	2	Yes

Step 5: Compute the estimate. Eight of the 10 trials were successes, so

P(\text{at least 2 makes}) \approx \frac{8}{10} = 0.8

Step 6: Report honestly. The estimate is about 0.8 based on only 10 trials. The true value is close to 0.648, so with more trials the estimate would likely drop. Ten trials is enough to practice the method but too few for a precise answer.

Practice questions

Which of the following is a valid probability model for the color of a randomly chosen marble from a bag?

$P(\text{red}) = 0.4$ , $P(\text{blue}) = 0.4$ , $P(\text{green}) = 0.3$
$P(\text{red}) = 0.5$ , $P(\text{blue}) = 0.2$ , $P(\text{green}) = 0.3$
$P(\text{red}) = 0.6$ , $P(\text{blue}) = -0.1$ , $P(\text{green}) = 0.5$
$P(\text{red}) = 0.3$ , $P(\text{blue}) = 0.3$ , $P(\text{green}) = 0.3$

Answer: $P(\text{red}) = 0.5$ , $P(\text{blue}) = 0.2$ , $P(\text{green}) = 0.3$

Check the sum in each case. The first gives

0.4 + 0.4 + 0.3 = 1.1

, too big. The third contains a negative probability, which is impossible, and also sums to 1.0 only if you ignore that. The fourth gives

0.9

, so an outcome is missing. Only the second sums to exactly

0.5 + 0.2 + 0.3 = 1

with every value between 0 and 1, so it is the valid model. Notice it is non-uniform, which is perfectly fine.

A spinner has 5 equal sectors: 2 red, 2 blue, 1 yellow. Write the probability model and state whether it is uniform.

Answer: $P(\text{red}) = \frac{2}{5}$ , $P(\text{blue}) = \frac{2}{5}$ , $P(\text{yellow}) = \frac{1}{5}$ ; the model is non-uniform.

The five sectors are the equally likely pieces, so each color's probability is its number of sectors over 5. Check the sum:

\frac{2}{5} + \frac{2}{5} + \frac{1}{5} = \frac{5}{5} = 1

, so the model is complete. It is non-uniform because yellow has probability

\frac{1}{5}

while red and blue each have

\frac{2}{5}

— the three color outcomes are not all equally likely, even though the five sectors are.

About 25 percent of the boxes of a cereal contain a sticker. Describe a simulation you could use to estimate the probability that a shopper who buys 4 boxes gets at least one sticker. Explain your device, your assignment, one trial, and what you would count.

Answer: Use a number cube is not ideal; instead use pairs of random digits 00 to 99 or a four-sector spinner. For example, spin a spinner with 4 equal sectors, one labeled 'sticker'. One trial is 4 spins (one per box). Count a trial as a success if at least one spin lands on 'sticker'. Run 50 trials and divide the number of successful trials by 50.

The key is matching 25 percent exactly. A four-sector spinner gives

\frac{1}{4} = 0.25

per box, which is a perfect match; random digits also work if you let 0 and 1 mean 'sticker' and 2 through 9 mean 'no sticker' — that gives 0.20, which is not 0.25, so that particular assignment would be wrong. A better digit version uses digits 1 through 4 for one box only after discarding, which is messy, so the spinner or a deck of four cards is cleaner. One trial must cover all 4 boxes because the question is about a shopper buying 4, and the denominator of the estimate is the number of trials, not the number of spins.

FAQ

How do I know whether a probability model is uniform?: Compare the probabilities of the outcomes you listed. If every listed outcome has the same probability, the model is uniform. If any two differ, it is non-uniform. Be careful: what looks like a neat list of outcomes is often non-uniform, as with the sums 2 through 12 from two number cubes.
Why must probabilities add to exactly 1?: Because your list of outcomes covers everything that can happen, and something must happen. A sum less than 1 means you left out an outcome; a sum greater than 1 usually means outcomes overlap or a probability is too large. Checking the sum is the fastest way to catch those errors.
How many trials should a simulation have?: More trials give a more reliable estimate. Twenty is a workable minimum for classwork, and 50 or 100 is much better if you can pool results with classmates. Decide the number before you start, and always report the number of trials with your estimate.
What if my simulation answer does not match the exact probability?: That is expected. A simulation estimates a probability, so results vary from run to run. Your design is wrong only if the device's probabilities do not match the real situation, if one trial does not model the whole event, or if you divided by the number of digits instead of the number of trials.

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